Finite Dimensional Global Attractor Research Articles

Long time dynamics of electroconvection in bounded domains

We discuss nonlinear nonlocal equations with fractional diffusion describing electroconvection phenomena in incompressible viscous fluids. We prove the global well-posedness, global regularity and long time dynamics of the model in bounded smooth domains with Dirichlet boundary conditions. We prove the existence and uniqueness of exponentially decaying in time solutions for H 1 H^1 initial data regardless of the fractional dissipative regularity. In the presence of time independent body forces in the fluid, we prove the existence of a compact finite dimensional global attractor. In the case of periodic boundary conditions, we prove that the unique smooth solution is globally analytic in time, and belongs to a Gevrey class of functions that depends on the dissipative regularity of the model.

Asymptotic Behavior and Numerical Simulations of a Conservative Phase-Field Model with Two Temperatures

One of the types of problem that has attracted the attention of mathematicians in recent years is the phase field system. The field of application includes materials science, where phenomena such as phase separation in alloys, crystal formation and thermal welding are legion. Among these phase transition systems, the family of conservative systems is very popular with industry. Indeed, minimising losses in production systems is a major issue for their profitability. In this paper, we study the well-posedness of the formulation and the asymptotic behaviour of the solutions, by proving the existence of a finite-dimensional global attractor for a conservative variant of the two-temperature phase field system with homogeneous Neumann boundary conditions. The inclusion of two temperatures in the definition of the enthalpy of the system is a necessity in the case of a non-simple material. In a simple material, once the phase-change temperature has been reached, the temperature of the system remains constant until the material has completely changed state. This is not true in the case of a non-simple material, where an increase in the temperature of the system is observed even after the phase-change temperature has been reached. To conclude the work, we present a method for numerically approximating the solution and carry out some numerical tests.

Synchronization phenomena in thermoelastic waves connected in parallel

In this paper, the phenomenon of synchronization in nonlinearly coupled thermoelastic waves is investigated. The existence of a finite-dimensional smooth global attractor A α is proven in the natural energy space, where α is the coupling parameter. It is also proved that the coupled thermoelastic wave system can be reduced to a single thermoelastic wave when α tends to infinity. Subsequently, the upper-semicontinuity of their attractors is proven as α → ∞ . These results are ultimately employed to establish the synchronization phenomena for coupled thermoelastic waves. To the best of our knowledge, the synchronization of thermoelastic waves has not been considered before.

Finite-dimensional global attractor for the three-dimensional viscous Camassa-Holm equations with fractional diffusion on bounded domains

Finite-dimensional global attractor for the three-dimensional viscous Camassa-Holm equations with fractional diffusion on bounded domains

WHAT HAS BIOLOGY DONE FOR MATHEMATICS: SUCCESS STORIES

Abstract. The long-time dynamics of ODE-PDE coupling models is a fascinating subject area. It is at very heart of understanding many important problems arising in the natural sciences. Heuristically, it is clear that the dynamics such coupled systems should depend drastically on the monotonicity properties of the ODE component. In the case, where this ODE component is "monotone", that is cannot produce internal stability (and all of the instability is driven by the coupling with the PDE component), one expects asymptotic compactness and the existence of finite-dimensional global attractor using the methods of infinite-dimensional dynamical systems. In contrast to that, in the case when ODE component is "nonmonotone", the ODE instability may produce asymptotic discontinuities and even may destroy the initially smooth spatial profile. Thus, in that case, the smoothing effect from the PDE component is not strong enough to suppress the development of discontinuities provided by the internal instabilities of the ODE component and as a result spatial discontinuities and extremely complicated spatial structures may appear.

Lamé system with weak damping and nonlinear time-varying delay

Abstract This article is concerned with the stability and dynamics for the weak damped Lamé system with nonlinear time-varying delay in a bounded domain. Under some appropriate assumptions, the global well-posedness and asymptotic stability are shown in the case where the delay coefficient is upper dominated by the damping one. Moreover, the finite dimensional global and exponential attractors have also been presented by relying on quasi-stability arguments. The results in this article is an extension of Ma, Mesquita, and Seminario-Huertas’s recent work [Smooth dynamics of weakly damped Lamé systems with delay, SIAM J. Math. Anal. 53 (2021), no. 4, 3759–3771].

Global attractors for a partially damped Timoshenko–Ehrenfest system without the hypothesis of equal wave speeds

This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system.

Dynamics of a class of extensible beams with degenerate and non-degenerate nonlocal damping

This work is concerned with new results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with three distinguished nonlocal nonlinear damping terms. In the first possibly degenerate case, the results feature the existence of a family of compact global attractors and a thickness estimate for their Kolmogorov’s $\varepsilon$-entropy. Then, in the non-degenerate context, the structure of the helpful nonlocal damping leads to the existence of finite-dimensional compact global and exponential attractors. Lastly, in a degenerate and critical framework, it is proved the existence of a bounded closed global attractor but not compact. To the proofs, we provide several new technical results by means of refined estimates that open up perspectives for a new branch of nonlinearly damped problems.

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.

Exponential attractors for modified Swift-Hohenberg equation in $\mathbb R^{N}$

A Cauchy problem for a modification of the Swift-Hohenberg equation in $\mathbb R^{N}$ with a mildly integrable potential is considered. Existence of exponential attractors containing a finite-dimensional global attractor in $H^2(\mathbb R^{N})$ is shown under the dissipative mechanism of fourth order parabolic equations in unbounded domains.

Asymptotic behavior of an Allen-Cahn type equation with temperature

In this paper, we are interested in the study of the asymptotic behavior of an Allen-Cahn type equation with temperature and endowed Dirichlet boundary conditions. Such a model has, in particular, applications in chemistry. In particular, we obtain well-posedness results and the existence of the finite-dimensional global attractor as well as the existence of exponentials attractors by means of classical arguments.

Existence and robustness results of attractors for partially-damped piezoelectric beams

This paper deals with a fully-dynamic piezoelectric beam model with a nonlinear force acting on the longitudinal displacements of the beam. The equations of motion follows a system of non-compactly coupled wave equation. As the magnetic effects are discarded by setting the magnetic permeability $ \mu $ to zero, the equations are fully decoupled to a single wave equation (electrostatic/quasi-static beam model). The existence of smooth finite-dimensional global attractors is proved by considering a damping injection to only one of the equations. Next, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Finally, to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, a singular limit problem $ \mu \to 0 $ is considered. This analysis allows to compare the attractors from one model to another in the sense of the upper semi-continuity of the attractor $ \mathcal{A}_{\mu} $ as $ \mu \to 0 $.

Robust exponential attractors for the Cahn-Hilliard-Oono-Navier-Stokes system

We focus in this article on a Cahn-Hilliard-Oono-Navier-Stokes system, which is a model consisting of a Navier-Stokes equation governing the fluid velocity coupled to the Cahn-Hilliard-Oono equation. We start our study by establishing the a priori estimates, the existence and uniqueness of the solution, and then the existence of a continuous dissipative semigroup generated by the system. Moreover, we prove the existence of the exponential attractors and, thus, of finite-dimensional global attractors. Finally, we construct a robust family of exponential attractors as a nonnegative parameter $ \alpha $ goes to zero.

Attractors for partially damped systems of binary mixtures of solids

This paper targets the long-time dynamics of a semilinear system modeling a binary mixture of solids with frictional dissipation mechanism acting only on the elastic equation and subjected to nonlinear source term. The coupling gives new contributions to the theory associated with nonlinear dynamics of partially damped semilinear systems of a binary mixture of solids. Using the quasi-stability methods, we prove the existence of a smooth finite-dimensional global attractor irrespective of the wave speeds of the system. Moreover, we establish the existence of exponential attractors.

Viral Infection Model with Diffusion and Distributed Delay: Finite-Dimensional Global Attractor.

We study a virus dynamics model with reaction-diffusion, logistic growth terms and a general non-linear infection rate functional response. The model has a distributed delay, including the case of state-selective delay. We construct a dynamical system in a Hilbert space and prove the existence of a finite-dimensional global attractor.

Finite fractal dimensional global attractor for abstract differential equations with state-dependent delay

We study the existence of a finite fractal dimensional global attractor for the multivalued semigroup generated by a general class of abstract differential equations with state-dependent delay. In the last section, some examples concerning partial differential equations with state-dependent delay are presented.

Uniform dynamics of partially damped semilinear Bresse systems

This paper is concerned with the Bresse system that arises in the modeling of arched beams. It is given by a system of three coupled wave equations that reduces to the well-known Timoshenko model when the arch curvature is zero. In a context of nonlinear elastic foundation, we establish the existence of smooth finite-dimensional global attractors, by adding dissipation mechanism in only one of its equations. In addition, we study the uniform boundedness of longtime dynamics with respect to the curvature parameter. These results have not been considered for partially damped semilinear Bresse or Timoshenko systems.

Dynamics of locally damped Timoshenko systems

In this paper, we study the long-time dynamics of a Timoshenko system modeling vibrations of beams with non-linear localized damping mechanisms acting on both displacement and angular rotation and subjected to non-linear source terms. Using recent quasi-stability methods, we prove the existence of smooth finite dimensional global attractor, which is characterized as unstable manifold of the set of stationary solutions. Moreover, the existence of exponential attractors is shown. These aspects were not previously considered for the Timoshenko system with localized damping.

Limiting dynamics in one-dimensional porous-elasticity

In this paper we investigate the long-term behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that the porous-elasticity problem converges to a quasi-static problem for the microvoids motion as a suitable parameter J tends to zero. Finite dimensional global attractor with additional regularity in J is obtained using the recent quasi-stability theory. Finally, we compare the porous-elasticity problem with quasi-static problem, in the sense of the upper-semicontinuity of their attractors as [Formula: see text].

Global and exponential attractors for mixtures of solids with Fourier’s law

This paper presents a study of the long-time dynamics of the dynamical system generated by a nonlinear system modeling mixture of solids with nonlinear damping and Fourier’s law. By using the recent quasi-stability theory, we prove the existence of a smooth finite dimensional global attractor, which is characterized as an unstable manifold of the set of stationary solutions. The quasi-stability of the system is achieved through an estimated stabilizability. Moreover, the existence of a generalized exponential attractor is shown.